A simple principle applies for what seems like a thorny issue: Nest punctuation that appears within punctuation by alternating punctuation marks to disambiguate–in this case, between double and single quotation marks.
One Level of Nesting
The most common reason for nesting punctuation is shown in section 1.3.7 of the MLA Handbook (p. 87): when you need to present a quotation within a quotation, use double quotation marks around the quotation incorporated into your text and single quotation marks around the quotation within that quotation:
In “Memories of West Street and Lepke,” Robert Lowell, a conscientious objector (or “C.O.”), recounts meeting a Jehovah’s Witness in prison: “‘Are you a C.O.?’ I asked a fellow jailbird. / ‘No,’ he answered, ‘I’m a J.W.’” (38-39).
Two Levels of Nesting
If the quotation enclosed in single marks also contains material–whether another quotation or the title of a work–that needs to be set off with quotation marks, use double quotation marks around that material. The pattern is double, single, double quotation marks. In other words, nest punctuation within punctuation and alternate to disambiguate:
“[Mr. Lawson] called out the name [Gogol] in a perfectly reasonable way, without pause, without doubt, without a suppressed smile, just as he had called out Brian and Erica and Tom. And then: ‘Well, we’re going to have to read “The Overcoat.” Either that or “The Nose”’” (Lahiri 89).
Lahiri, Jhumpa. The Namesake. Mariner Books, 2004.
A Related Case
The same principle applies when you need to incorporate parenthetical material. Alternate between parentheses and brackets, as in this aside:
(Early in The Namesake, the narrator explains that “[t]hough Gogol doesn’t know it, even Nikolai Gogol renamed himself. . . . [He had also published under the name Yanov, and once signed his work ‘OOOO’ in honor of the four o’s in his full name]” [Lahiri 97]).